HEMT (4 K)
5.3 General properties of the excess noise
5.3.1 Pure phase (frequency) noise
Frequency (Hz)
N o is e P S D (d B c/ H z) R o ta ti o n a n g le (d eg )
10
010
110
210
310
410
560 90 120 150
-100 -90 -80 -70 -60 -50 -40
Figure 5.3: Noise spectra in the phase (Saa(ν), solid line) and amplitude (Sbb(ν), dashed line) directions, and the rotation angle (φ(ν), dotted line). The noise data are from the same Nb/Si resonator under the same condition as in Fig. 5.2.
A typical pair of spectraSaa(ν) andSbb(ν) are shown in Fig. 5.3, along with the rotation angle φ(ν), defined as the angle between va(ν) and theI axis. Three remarkable features are found for
all noise data. First, φ(ν) is independent of ν within the resonator bandwidth (the r.m.s. scatter isσφ ≤0.4◦ per 10 Hz frequency bin from 1 Hz to 5 kHz in Fig. 5.3), which means that only two special directions, va and vb, diagonalizeS(ν). Second,va is always tangent to the IQ resonance circle while vb is always normal to the circle, even when f is detuned from fr. Because Saa(ν) and Sbb(ν) are the noise spectra projected into these two constant directions according to Eq. 5.3, they are equal to the voltage noise spectra calculated from Fourier transform of the projected time domain noise dataδξk(t) andδξ⊥(t),
Saa(ν) =Sk(ν)
Sbb(ν) =S⊥(ν). (5.8)
Third,Saa(ν) is well aboveSbb(ν) (see Fig. 5.3). When we plot the noise ellipse according to Eq. 5.4 and 5.5 using a integration bandpassν1= 1 Hz andν2= 1 kHz, we find the major axis of the noise ellipse is always in the phase direction, and the ratio of the two axes is always very large (8 for the noise ellipse shown in Fig. 5.2(b)).
Fig. 5.3 also shows that the amplitude noise spectrum is flat except for a 1/νknee at low frequency contributed by the electronics. The amplitude noise is independent of whether the synthesizer is tuned on or off the resonance, and is consistent with the noise temperature of the HEMT amplifier.
The phase noise spectrum has a 1/ν slope below 10 Hz, typically a ν−1/2 slope above 10 Hz, and a roll-off at the resonator bandwidth fr/2Qr (as is the case in Fig. 5.3) or at the intrinsic noise bandwidth ∆νn, whichever comes first. The phase noise is well above the HEMT noise, usually by two or three orders of magnitude (in rad2/Hz) at low frequencies. It is well in excess of the synthesizer phase noise contribution or the readout system noise.
Quasi-particle fluctuations in the superconductor, perhaps produced by temperature variations or the absorption of high frequency radiation, can be securely ruled out as the source of the measured noise by considering the direction in the IQ plane that would correspond to a change in quasi-particle densityδnqp. According to the discussion in Section 2.4, both the real and inductive parts of the complex conductivityσrespond linearly toδnqp,δσ=δσ1−iδσ2=κ|σ|δnqp, resulting in a trajectory that is always at a nonzero angle ψ= tan−1(δσ1/δσ2) to the resonance circle, as indicated by the dashed lines in Fig. 5.2(a) and (b). Mattis-Bardeen calculations yieldψ= tan−1[Re(κ)/Im(κ)]>7◦ for Nb below 1 K, so quasi-particle fluctuations are strongly excluded, sinceψ >> σφ. Furthermore, ψ is measured experimentally by examining the response to X-ray, optical/UV, or submillimeter photons, and is typicallyψ≈15◦ ([24, 50], and see Section 5.6.3).
100 101 102 103 104 105 10−10
10− 9 10− 8 10− 7 10− 6
-99 dBm -95 dBm -91 dBm -87 dBm
Frequency (Hz)
Voltage noise PSD (V /Hz)2
(a)
Fractionalfrequencynoise(1/Hz)
Frequency (Hz) Phasenoise(rad2/Hz)
100 101 102 103 104 105 10−9
10−8 10−7 10−6 10−5
10−21 10−20 10−19 10−18 10−17 10−16
-99 dBm -95 dBm -91 dBm -87 dBm
(b)
Figure 5.4: Excess noise in the phase direction under different readout powers Pµw. (a) Voltage noise spectrum Sk(ν). (b) Phase noise spectrum Sδθ(ν) (left axis) and fractional frequency noise spectrumSδfr(ν)/fr2(right axis). The readout powers of the 4 curves arePµw=-87 dBm, -91 dBm, -95 dBm, -99 dBm from top to bottom in (a) and from bottom to top in (b). The data is measured from a 200 nm thick Al on sapphire resonator.
5.3.2 Power dependence
The excess noise has a dependence on the microwave readout power Pµw. Fig. 5.4 compares the measured noise spectra of a resonator under four different readout powers in steps of 4 dBm. We found the voltage noise increases with the readout power, as shown in Fig. 5.4(a). A 2 dB separation is found between the two adjacent noise spectra, suggesting
Sk(ν)∝Pµw12 . (5.9)
The excess noise, when converted to phase noise or frequency noise, decreases with readout power.
The same separation of 2 dB but with a reversed order (top curve with the lowestPµw) is seen in Fig. 5.4(b), which suggests:
Sδθ(ν)∝Pµw−12, Sδfr(ν)/fr2∝Pµw−12. (5.10) Eq. 5.9 and Eq. 5.10 are consistent because the radius of the resonance looprscales asr∝Pµw12 .
To compare the excess noise among resonators with different fr and Qr, we plot the frequency noiseSδfr(ν)/fr2 as a function of the microwave power inside the resonator (the internal power). It can be shown that the internal powerPint is related to the readout powerPµw by
Pint= 2 π
Q2r Qc
Pµw (5.11)
for a quarter-wave resonator.
The frequency noise vs. internal power for resonators with different fr and Qr on the same
Internal Power (dBm)
S(1kHz)/f(1/Hz)
-54 -52 -50 -48 -46 -44 -42 -40 -38 -36 10
-1910
-18δfrr 2
Figure 5.5: Frequency noise at 1 kHzSδfr(1 kHz)/fr2vs. internal powerPint falls on to straight lines of slope -1/2 in the log-log plot indicating a power law dependance: Sδfr/fr ∝Pint−1/2. Data points marked with“+”,“”, and “*” indicate the on-resonance (f =fr) noise of three different resonators (with different fr and Qr on the same chip) under four different Pµw. Data points marked with
“◦” indicate the noise of resonator 1 (marked with “*”) measured at half-bandwidth away from the resonance frequency (f =fr±fr/2Qr) under the same fourPµw. The data is measured from a 200 nm thick Al on sapphire device.
substrate are compared in Fig. 5.5. The data points, from three different resonators, four different readout powers, driven on-resonance and detuned, fall nicely onto a straight line of slope -1/2 in the log-log plot, suggesting that the frequency noise depends on the internal powerPintof the resonator by a power law
Sδfr(ν)/fr2∝Pint−12. (5.12) The power law index -1/2 in Eq. 5.12 is suggestive. For comparison, amplifier phase noise is a multiplicative effect that would give a constant noise level independent ofPint, while the amplifier noise temperature is an additive effect that would produce a 1/Pint dependence.
5.3.3 Metal-substrate dependence
The excess noise also depends on the materials used for the resonator. In Fig. 5.6, we plot the frequency noise spectrum at 1 KHz Sδfr(1 kHz)/fr2 against internal powerPint for five resonators made of different metal-substrate combinations (all substrates used are crystalline substrates). In addition to the power dependence Sδfr(ν)/fr2 ∝Pint−1/2, we find that the noise levels are material dependent. In general, sapphire substrates give lower phase noise than Si or Ge, roughly by an order of magnitude in the noise power. However the Nb/Si resonator showed low noise comparable with Al/sapphire resonator, suggesting that the etching or interface chemistry, which is different for Nb and Al, may play a role. Two Al/Si resonators with very different Al thicknesses and kinetic
inductance fractions[64] fall onto the dashed equal-noise scaling line, strongly suggesting that the superconductor is not responsible for the phase noise.
320 nm A l on Si 40 nm A l on Si 200 nm N b on Si 200 nm A l on Sapphire 200 nm A l on Ge
Internal Power (dBm) S
δfr(1 k H z )/ f
2 r(1 /H z )
-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 10
−2010
−1910
−1810
−17Figure 5.6: Power and material dependence of the frequency noise atν = 1 kHz. All the resonators shown in this plot havew=3µm,g=2µm and are measured around 120 mK. The spectra used in this plot are single-sided (ν >0).
As will be discussed in great detail later in this chapter, the TLS on the surface of the resonator, either metal surface or the exposed substrate surface, are responsible for the excess noise. Therefore, the metal-substrate dependence of the excess noise shown in Fig. 5.6 turns out to have nothing to do with the bulk properties of the superconductor or the substrate. Instead, it’s their surface or interface properties that make a difference. For example, the metal Al, Nb and crystalline Si, Ge can all form a native oxide layer on the surface, which can be the host material of the TLS. The defects, impurities and chemical residues introduced during etching and other processes of the fabrication may be another source of TLS.
5.3.4 Temperature dependence
The temperature dependence of the excess frequency noise is best demonstrated by the experiment in which the noise of a Nb on Si resonator is measured at temperatures below 1 K. Because Nb has a transition temperatureTc = 9.2 K, the noise contribution from superconductor are frozen at T <1 K. Any temperature dependence of noise has to be from other low energy excitations — TLS in the resonator.
Fig. 5.7 shows the measured phase and amplitude noise spectra under readout power Pµw =
−85 dBm at several temperatures between 120 mK and 1200 mK. While the amplitude noise (S⊥(ν), in green) remains almost unchanged, the phase noise (Sk(ν), in blue) decreases steeply with temper- ature. As mentioned earlier, the amplitude noise spectrumS⊥(ν) is consistent with the noise floor
100 101 102 103 104 105 10−4
10−2 100 102
Voltage noise PSD [ADU2 /Hz]
Frequency [Hz]
S||
S⊥ S|| − S⊥
Figure 5.7: Phase noise (Sk(ν), blue curves) and amplitude noise (S⊥(ν), green curves) spectra measured at T=120, 240, 400, 520, 640, 760, 880, 1000, 1120 mK (from top to bottom). The true phase noise can be calculated by subtracting the amplitude noise from the phase noise, which is plotted as the red curves. The voltage unit used here is the unit of our AD card with 1 V = 32767 ADU. The data is measured from a 200 nm Nb on Si resonator under a fixed readout power Pµw=−85 dBm.
of the readout electronics (mainly limited by the noise temperature of our HEMT). Therefore, we calculate the “true” phase noise by subtracting the measuredS⊥(ν) fromSk(ν) and the results are plotted in red curves in Fig. 5.7.
To better quantify the temperature and power dependence of the frequency noise, we retrieve the noise values at 1 kHz from the phase noise spectrum (red curve) at each readout power and each temperature. The 1 kHz frequency noise Sδfr(1 kHz)/fr2 is plotted as a function ofPµw and T in Fig. 5.8. The even spacing (∼ 2 dB) between any two adjacent noise curves indicates the Pint−1/2 dependence of frequency noise as expected. At a fixed Pµw, we find the frequency noise roughly falls onto a power-law relationship and at intermediate temperatures 300 mK< T <900 mK the temperature dependence is close to
Sδfr(1 kHz)/fr2∝T−2 (5.13)
as indicated by the parallel solid lines in Fig. 5.8. This scaling is consistent with theT−1.73scaling found by Kumar[63], where he was fitting for a broader range of temperatures.
In addition to the noise, the resonance frequency fr and quality factor Qr also show strong temperature dependence, which are shown in Fig. 5.9. Later in this chapter we will see plenty examples of similarfr(T) andQr(T) curves and show that they can be well explained by the TLS theory.
102 103 10−22
10−21 10−20 10−19 10−18
T [mK]
S δ f r(1 kHz)/f r2 [1/Hz]
T
−2Figure 5.8: Frequency noise atν= 1 kHz as a function of temperature under several readout powers.
The readout powersPµware from -105 dBm to -73 dBm in step of 4 dBm from top to bottom. The solid lines indicate T−2 temperature dependence. The data is measured from a 200 nm Nb on Si resonator.
In summary, the measured temperature dependence of resonance frequency, quality factor, and frequency noise strongly suggest to us that TLS in the dielectric materials are responsible for the noise.
5.3.5 Geometry dependence
The geometry dependence of the frequency noise was carefully studied with a Nb on sapphire geometry-test device, which contains CPW resonators with five different center strip widths (sr
=3µm, 5µm, 10µm, 20µm, and 50µm) and with the ratio between the center strip width and the gap width fixed to 3:2. Here we only present the conclusions, while leaving the detailed data and analysis to Section 5.5.2.2, after the introduction of TLS theory and a semi-empirical noise model.
Fig. 5.10 shows the measured frequency noise (before and after the correction for coupler’s noise contribution) at ν = 2 kHz as a function of center strip widthsr under the same internal power Pint=−25 dBm. We find that the frequency noise has a geometrical scaling
Sδfr(ν)/fr2∝1/s1.6r . (5.14)
The noise data as well as the temperature-dependentfr(T) andQr(T) data measured from this geometry-test device will be discussed in great detail in Section 5.5.2.2. As we will show there, these data not only confirm the TLS hypothesis but further point to a surface distribution of TLS and rule out a uniform distribution of TLS in the bulk substrate.
0 200 400 600 800 1000 1200 4.3472
4.3473 4.3473 4.3473 4.3473 4.3473
T [mK]
f r [GHz]
200 400 600 800 1000 1200 1.5
2 2.5 3 3.5 4 4.5x 105
T [mK]
Q
Figure 5.9: Resonance frequency (a) and quality factor (b) as a function of temperature under several readout powers. The readout powersPµw are from -105 dBm to -73 dBm in steps of 4 dBm from bottom to top in both plots. The data is measured from a 200 nm Nb on Si resonator.